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We study a class of high order enriched unfitted finite element methods or generalized finite element methods (GFEM) to solve a larger class of interface problems i.e., one-dimensional parabolic interface problems with discontinuous solutions including those having implicit or Robin-type interface conditions. The difficulty of the development of the enriched finite element method is to build enrichment functions that capture the imposed discontinuity of the solutions without affecting the condition number from fast growth. Recently, the high order stable generalized finite element method (SGFEM) of Chou et.al. [5], [15] was developed using two simple discontinuous one-sided enrichment functions. We apply the same technique in these types of parabolic interface problems with discontinuous solutions. Also, we study the flux recovery via post-processing and investigate the superconvergence behavior of the high order SGFEM. The flux recovery is done at nodes and interface points first and by interpolation at the remaining points. Next, we derive practical optimal order estimates for the high order methods with time-continuous and full discretization. To prove the efficiency of the high order SGFEM, we test our methods on a multi-layer wall model for drug-eluting stents.Additionally, we extend the high order enriched finite element method to the multi-layer porous wall model, and under some realistic assumptions on the initial data and body force, we study the long-time error stability of the solution.